Understanding the rapidity dependence of v 2 and HBT at RHIC

1. Understanding the rapidity dependence of v2 and HBT at RHIC M. Csanád (Eötvös University, Budapest) WPCF 2005 August 15-17, Kromeriz

2. PLB505:64-70,2001hep-ph/0012127 Hydro equations + EoS Self-similar solution: Source S(x,p) Phase-space distribution Boltzmann-equation PRC67:034904,2003hep-ph/0108067 PRC54:1390-1403,1996hep-ph/9509213 Observables N1(p), C2(p1,p2), v2(p) How analytic hydro works Scheme works also backwords* *For a certain time-interval

3. cs2 = 2/3 cs2 = 1/3 Sensitivity to the EoS • Different initial conditions, different equation of statebut exactly the same hadronic final state possible. (!!) • This is an exact, analytic result in hydro( !!)

4. Buda-Lund hydro • 3D expansion, symmetry • Local thermal equilibrium • Analytic expressions for the observables (no numerical simulations, but formulas) • Reproduces known exact hydro solutions (nonrelativistic, Hubble, Bjorken limit) • Core-halo picture

5. Time dependence • Blastwave or Cracow model type of cooling vs Buda-Lund cooling, cs2= 2/3, half freeze-out time see: http://csanad.web.elte.hu/phys/3danim/

6. A useful analogy • Core  Sun • HaloSolar wind • T0,RHIC  T0,SUN  16 million K • Tsurface,RHIC  Tsurface,SUN  6000 K • RG Geometrical size • t0 Radiation lifetime • <bt>  Radial flow of surface (~0) • DhLongitudinal expansion (~0) Fireball at RHICFireball Sun

7. Buda-Lund in spectra, HBT… J.Phys.G30: S1079-S1082, 2004 nucl-th/0403074

8. Main axes of expanding ellipsoid: • 3D expansion, 3 expansion rates: • Introducing space-time eccentricity: • Hubble type of expansion: • Aprroximation: • Additionally: Ellipsoidal generalization • Axially symmetric case: RG, ut

9. Generalized Cooper-Frye prefactor: • Proper-time distribution: • Temperature-distribution: The ellipsoidal Buda-Lund model • The original model was developed for axial symmetry  central collisions • In the most general hydrodynamical form • (‘Inspired by’ nonrelativistic solutions): • Fugacity: • Shape of distributions: • Four-velocity distribution: Hubble-flow M.Cs., T.Csörgő, B. Lörstad: Nucl.Phys.A742:80-94,2004; nucl-th/0310040

10. Core-halo picture: • One-particle spectrum with core-halo correction: • Two-particle correlation function: • Width of it are the HBT radii Observables from BL hydro • Flow coefficients:

11. Temperature gradient Expansion rate Scaling variable Space-time rapidity of the point of maximal emittivity HBT radii ‘Harmonic sum’ of geometrical and thermal radii

12. HBT(mt,φ,h) Rout Rside Rlong • Dramatic change at low mt • No change at high mt • Radii decreasing with increasing mt or y

13. Hydro scaling in HBT • Radii depend on mt and y through mt

14. Hydro scaling in HBT • Radii depend on mt and y through mt

15. Hydro scaling in HBT • Radii depend on mt and y through mt

16. The elliptic flow • One-particle spectrum: • Pseudorapidity dependence mostly not understood (except see Hama/SPHERIO) • The m-th Fourier component is the m-th flow • Depends on pseudorapidity and transverse momentum

17. At large pseudorapidities… • If the point of maximal emittivity (saddlepoints) is near the longitudinal axis: and , introducing • Here hs is the space-time rapidipy of the saddlepoint • , and so • Rapidity grows  the asymmetry vanishes(saddlepoint goes to the z axis)  elliptic flow vanishes

18. Hydro predicts scaling • Scaling variable • For every type of measurement: • Elliptic flow depends on every physical parameter only through w • Scaling curve I1 / I0?

19. Fits to PHOBOS data

20. Fit parameters • Fitted parameters: eccentricity & Dh • Fixed (non-essential) parameters (from spectra and HBT fits):

21. Error contours

22. Universal scaling • Scale parameter w The perfect fluid extends from very small to very large rapidities at RHIC

23. Conclusions I. • Buda-Lund model describes HBT data @RHIC • Predictedion of the rapidity dependence of HBT radii • Hydro scaling present in HBT radii? Straightforward to check!

24. Conclusions II. • Buda-Lund model describes v2(h) data @RHIC • The vanishing elliptic flow at large h: Hubble flow + finite longitudinal size • v2(h) data (2005) collapse to the theoretically predicted (2003) scaling function of • The perfect fluid is present in AuAu in the whole h space

25. Thanks for your attention Spare slides coming …

26. Nonrelativistic hydrodynamics • Equations of nonrelativistic hydro: • Not closed, EoS needed: • We use the following scaling variable: • X, Y and Z are characteristic scales, depend on (proper-) time

27. A nonrelativistic solution • A general group of scale-invariant solutions (hep-ph/0111139): • This is a solution, if the scales fulfill: •  (s) is arbitrary, e.g.  constant   gaussian, or: Buda-Lund Bondorf-Zimanyi-Garpman

28. Some numeric results from hydro • Propagate the hydro solution in time numerically:

29. A relativistic solution • Relativistic hydro: with • A general group of solutions (nucl-th/0306004): • Overcomes two shortcomings of Bjorken’s solution: • Rapidity distribution • Transverse flow • Hubble flow  lack of acceleration

30. The emission function • The phase-space distribution looks like Maxwell-Boltzman,for sake of simplicity with the constant: • Consider the collisionless Boltzmann-equation Calculates the source of a given phase-space distribution: • Emission function in the simplest case (instant. source, at t=t0):